Minnesota Symposia on Child Psychology, Volume 38nunez/web/Nunez_ch3_MN.pdf · 2017-04-13 · functioning in corresponding brain areas was momentarily disrupted (Göbeletal.,2006).Moreover,forseveraldecades,lesionstudieshave

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CHAPTER

3

How Much Mathematics Is“Hardwired,” If Any at All

Biological Evolution, Development, and theEssential Role of Culture

Rafael Núñez

INTRODUCTION

The investigation of mathematical cognition in developmental psychol-ogy has been primarily confined to the study of numerical cognition—basic facts involving quantity, such as the acquisition of number wordsand some competences for numerical estimation and calculation. Butmathematics is much more than simple counting numbers and basicarithmetic. Mathematics, seen by many as the queen of sciences, is aunique and peculiar body of knowledge. It is objective, precise, relational,stable, robust, consistent, and highly effective in modeling aspectsof the world—a feature expressed a century ago by the Hungarianphysicist Eugene Wigner as the “unreasonable effectiveness of mathe-matics” (Wigner, 1960). Importantly, mathematics is quintessentiallyabstract—the very entities that constitute this domain are idealized

83Minnesota Symposium on Child Psychology: Culture and Developmental Systems, Volume 38,Maria D. Sera, Michael Maratsos and Stephanie M. Carlson© 2017 by John Wiley & Sons, Inc. All rights reserved. Published by John Wiley & Sons, Inc.

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84 Minnesota Symposium on Child Psychology

mental abstractions that cannot be observed directly through the senses.A Euclidean point, for instance, is dimensionless and cannot be actuallyperceived. Even, say, the number “eight” cannot be observed directly, aswe might perceive a collection we label as having eight glasses or eightchairs, but we do not perceive the number “eight” as such. Moreover,mathematics is a conceptual system that relies almost entirely on writtenpractices—notations and symbols that make it happen. Cultures withno writing traditions or record-keeping practices may have some basiccounting systems, but beyond that they have virtually no mathematics.Finally, mathematics is special because, after all, it is a strange nonintu-itive conceptual system that often contains less than obvious concoctedfacts that are taught dogmatically and that usually remain unexplained.For instance, why does the multiplication of two negative numbers yielda positive result? Or why is the empty set a subset of every set? Or if 2! =1 × 2, 3! = 1 × 2 × 3, and so on, why then is 0! = 1? None of these math-ematical facts seems to find an instantiation in the natural world, and yetmathematics is full of these purely imaginary (and seemingly arbitrary)“truths” that, through formal definitions, axioms, and algorithms, providestability and consistency throughout its conceptual edifice.

So, what kind of thing is mathematics then? What makes it possible?Can the study of the nature of mathematics inform research in devel-opmental psychology? Is there any core of mathematics that is, as someneuroscientists and developmental psychologists proclaim, “hardwired”in the human brain? In this chapter, I address these questions and analyzethe claim that humans are biologically endowed with some kind ofprotomathematical apparatus. To make the enterprise tractable withinthe confines of this chapter, I focus on the mathematical concept of thenumber line, as it provides an excellent case study for the investigationof these matters. The number line—a conceptual tool that allows fornumbers to be conceived as locations along a line mapping numericaldifferences onto differences in spatial extension, as they appear in a ruler(technical details will be given below in the section entitled “Aren’tNumber-to-Space Mappings ‘Hardwired’?”)—is mathematically simple,yet it is extraordinarily powerful, and it is cognitively sophisticatedenough to provide insight into the issues at stake. Prototypically, the

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How Much Mathematics Is “Hardwired,” If Any at All 85

number line has a linear mapping (i.e., with regular “steps” between suc-cessive whole numbers), but it could also have a nonlinear mapping as in aslide rule, which, having a logarithmic mapping, exhibits “steps” betweennumbers that get compressed as the number increase in magnitude. Ievaluate the widespread assumption in contemporary research in thepsychology and neuroscience of numerical cognition that the mental rep-resentation of the number line (linear or logarithmic)—themental numberline—is somewhat innate, “hardwired” in the human brain. More specif-ically, I analyze two central underlying tenets of this nativist position:(1) that the representation of number is inherently spatial (as a line is)and (2) that the number-to-space mapping—essential for the numberline concept—is a universal intuition rooted directly in the brain throughbiological evolution, independent of culture, language, and education.

I begin by describing the major ideas involved in these nativist claims,pointing to some of the problems involved. I then review materialfrom the history of mathematics, in particular from Old Babylonianmathematics, and from specific developments in 17th-century Europeanmathematics, that is at odds with the nativist position. I follow thehistorical analysis with a review of results from recent empirical studies inexperimental psychology as well as in cross-cultural cognition conductedwith the Mundurukú from the Amazon and the Yupno from Papua NewGuinea. These results show that the representation of number is notinherently spatial and that the intuition of mapping numbers to space isnot universal, thus challenging the nativist claims. Taking the historicalobservations and experimental results together, I argue that even an ideaas fundamental as number-to-space mapping—let alone the technicalconcept of number line or any more advanced mathematical concept—isfar from having evolved via natural selection, requiring substantialcultural mediation and scaffolding for it to occur. I close the chapterwith a discussion on the problems brought by the teleological structure ofthe nativist arguments, which occlude the constitutive role that cultureplays in mathematical concepts. Using an analogy taken from culturallygenerated and sustained motor activity—snowboarding—I analyze howthe investigation of the origin and nature of the concept of the numberline shows the essential role of culture in high-level cognition and howthis informs the understanding of the interplay between cultural processesand biological mechanisms in developmental theory.

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NATIVISM IN COGNITIVE DEVELOPMENT,COGNITIVE NEUROSCIENCE, AND ANIMALCOGNITION

There is a widespread view, inspired by nativist ideas, that sees the originof mathematical entities in biological evolution, existing independentlyof cultural and linguistic dynamics. This quote expresses the positionclearly: “Mathematical objects may find their ultimate origin in basicintuitions of space, time, and number that have been internalized throughmillions of years of evolution in a structured environment and that emergeearly in ontogeny, independent of education” (Dehaene, Izard, Spelke, &Pica, 2008a, p. 1217). From the same perspective, and with respect to thenumber line, specific claims have been made in cognitive neuroscience:“Different studies reporting introspective descriptions, behavioral data,and neuroimaging data support the assumption of a continuous, analogi-cal, and left-to-right-oriented mental number line representing numbers,which is localized in the intraparietal sulcus” (Priftis, Zorzi, Meneghello,Marenzi, & Umiltà, 2006, p. 680). Not only is the mental line said to bedirectly “represented” in the brain without cultural, linguistic, or educa-tional influences; these authors go further to make specific anatomicalclaims about where exactly and in what “direction” the line is represented.

Similar positions are defended in cognitive development and even inanimal cognition.With respect to themore fundamental number-to-spacemappings, a line of work has claimed that such mappings are manifestedspontaneously in human infants and young children, as proclaimedby papers with titles reading “Spontaneous Mapping of Numberand Space in Adults and Young Children” (de Hevia & Spelke, 2009) and“Number-Space Mapping in Human Infants” (de Hevia & Spelke,2010). And, going phylogenetically much farther, recent titles in animalcognition reports have announced “Rhesus Monkeys Map Number ontoSpace” (Drucker&Brannon, 2014) and even “Number-SpaceMapping inthe Newborn Chick Resembles Humans’ Mental Number Line” (Rugani,Vallortigara, Priftis, & Regolin, 2015). What is the rationale of thesenativist claims, and why do they look so appealing? And, importantly, dothe data support these claims? (This latter question is addressed in thesection entitled “Aren’t Number-to-SpaceMappings ‘Hardwired’?” p. 90.)

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For a start, there is abundant research suggesting that space might bea natural grounding for number representation. After all, space providescountless metaphors for number that pervade the history of modern math-ematics (Lakoff & Núñez, 2000). And, more specifically, supported bydistinct spatial brain areas (Feigenson, Dehaene, & Spelke, 2004; Göbel,Calabria, Farnè, & Rossetti, 2006; Zorzi, Priftis, & Umiltà, 2002), spaceplays a major role in number processing (Bächtold, Baumüller, & Brugger,1998; Dehaene, Bossini, &Giraux, 1993; Gevers, Reynvoet, & Fias, 2003;for ameta-analysis, seeWood,Willmes,Nuerk,&Fischer, 2008).Crucially,linear space is said to be readily employed by children (Booth & Siegler,2006; Siegler&Booth, 2004) when confronted with the number-line task,in which they are asked to report numerical estimations on a line segment.(Wecomeback to this later in the subsection entitled “AreNumberMentalRepresentations Inherently Spatial?”, p. 100.)

The tradition of studying number–space relationships is in fact long.Consistent with descriptions already made by Galton in 1880, thatnumbers are pictured on a line, behavioral studies in the 1990s haveshown that there are associations between spatial-numerical responsecodes (SNARC), which appear to have a specific left–right horizontalspatial orientation (in people who grow up in cultures with left-to-rightwriting traditions): Western participants are faster to respond to largenumbers with the right hand and to small numbers with the left hand(Dehaene et al., 1993). Such results have led to the claim “that thecore semantic representation of numerical quantity can be linked to aninternal ‘number line,’ a quasi-spatial representation on which numbersare organized by their proximity” (Dehaene, Piazza, Pinel, & Cohen,2003, p. 498). And neuropsychological studies with unilateral neglectpatients (with right parietal lesions) appear to support this idea. Whenasked to bisect a line segment, these patients tend to locate the middlepoint farther to the right (ignoring the left space). When asked tofind the middle of two orally presented numbers, they answer withnumbers greater than the correct answer (Priftis et al., 2006; Zorzi et al.,2002). Similar responses were observed in healthy participants whenfunctioning in corresponding brain areas was momentarily disrupted(Göbel et al., 2006). Moreover, for several decades, lesion studies haverevealed the involvement of the parietal cortex in number processing

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(Gerstmann, 1940) and the systematic activation of the parietal lobesduring calculation (Roland & Friberg, 1985), facts that have been con-firmed with positron emission tomography (PET) (Dehaene et al., 1996)and functional magnetic resonance imaging (fMRI) studies (Rueckertet al., 1996). Taken together, these findings have led to the proposal thatthe parietal lobes contribute to the representation of numerical quantityon a mental number line (Dehaene et al., 2003; Dehaene & Cohen,1995). Consistent with these claims, the mental number line is routinelyinvoked when interpreting a variety of results in numerical cognition,from early approximate numerosity discrimination abilities (Feigensonet al., 2004) to simple “subtractions” in preverbal infants and nonhumanprimates (Dehaene et al., 2003), to lateralized behavioral biases innewborn chicks driven by brain asymmetries (Rugani et al., 2015). Asa result, generalized statements that “to compare numeric quantities,humansmake use of a ‘mental number line’ with smaller quantities locatedto the left of larger ones” (Doricchi, Guariglia, Gasparini, & Tomaiuolo,2005, p. 1663; emphasis added) or even that, in the animal (tetrapod)kingdom at large, “spatial mapping of numbers from left to right may bea universal cognitive strategy available soon after birth” (Rugani et al.,2015, p. 536) are taken for granted and go largely unchallenged.

In sum, the central idea in these nativist positions in cognitive neuro-science, cognitive development, and animal cognition is that there is a“core” in human cognition that is mathematical per se (the mental num-ber line, in this case): a sort of mathematical embryo, which in itself isculture-, language-, and education-free. At this point it is therefore advis-able to ask—leaving the number line and its underlying number-to-spacemappings aside—whether with respect to quantity alone, there are any“hardwired” capacities at all.

QUANTITY-RELATED “HARDWIRED” CAPACITIES?YES,…ARE THEY MATHEMATICAL? NO

Quantity-related capacities have been empirically investigated for morethan half a century. Experimental psychologists in the late 1940s began todocument the capacity for making quick, error-free, and precise judgmentsof the quantity of items in arrays of up to three or four items. They called

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this capacity subitizing, from the Latin word for “sudden” (Kaufmann, Lord,Reese, & Volkmann, 1949). Since these experimental findings many stud-ies have shown that humans (and other species) possess a sense of quantityobservable at an early age, before a demonstrable influence of languageand schooling. For instance, at 3 or 4 days, a baby is able to discriminatebetween collections of two and three items (Antell & Keating, 1983) andbetween sounds of two or three syllables (Bijeljac-Babic, Bertoncini, &Mehler, 1991). Under certain conditions, they can distinguish three itemsfrom four (Strauss & Curtis, 1981; van Loosbroek & Smitsman, 1990). By4.5 months, babies exhibit behaviors that some have interpreted as hav-ing a rudimentary understanding of elementary arithmetic as in “one plusone is two” and “two minus one is one” (Wynn, 1992). And as early as6 months, infants are able to discriminate between large collections ofobjects on the basis of approximate quantity, provided that they differ bya large ratio (8 versus 16 but not 8 versus 12; Xu & Spelke, 2000).

In sum, there is solid evidence showing that humans, and manynonhuman animals (Mandler & Shebo, 1982), do indeed possess somequantity-related innate capacities involving numerosity discrimination.Within a small numerosity range they are precise, as in subitizing, andapproximate, as in the case of large numerosity comparison. But thecrucial question is: Are these numerosity-related capacities mathematical?Do these phenomena constitute mathematics as such? If not mathematicalproper, it is indeed tempting to call these capacities protomathematical,or early-mathematical, or premathematical. But doing so has the problemof invoking profoundly misleading teleological arguments, which Ianalyze at the end of this chapter. For the moment, suffice to say thatnumerosity discrimination has been found even in mosquitofish, whichhas led scholars in animal cognition to interpret results as “spontaneousnumber representation” in fish (Dadda, Piffer, Agrillo, & Bisazza, 2009).Such moves bring the risk of ascribing “numerical” properties likeprecision, order, compositionality, or operativity to far simpler abilitiesthat only involve stimuli discrimination, paving the way for a teleo-logical argument that thousands of species, from fish to humans, have“number representations” as a result of biological evolution. Whetherto call these capacities mathematical is not a mere harmless superficialsemantic matter, as doing so brings important (misguided) theoretical

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consequences. As mentioned, mathematics is, among others, precise(not just approximative), relational, combinatorial, consistent, symbolicand abstract. Numerosity-related capacities in themselves do not possesthese properties: they are not mathematical as such. Subitizing andlarge (approximate) numerosity discrimination seem to be innate, andthey may provide necessary early cognitive preconditions (not precursors!)for numerical abilities (Núñez, 2009), but they are not numerical (ormathematical) as such, in the same way that an infant’s first walking stepsmay provide necessary early motor preconditions for snowboarding abilities,but they are not about snowboarding as such (more on this at the end ofthe chapter).

AREN’T NUMBER-TO-SPACE MAPPINGS“HARDWIRED”? NO

As mentioned, recent reports in developmental psychology and animalcognition have claimed that number–spacemappings aremanifested spon-taneously in human infants and young children without the influence ofculture (de Hevia & Spelke, 2009, 2010). And, going farther, that suchmappings manifest even in monkeys (Drucker & Brannon, 2014) andnewborn domestic chicks (Rugani et al., 2015). Wouldn’t this constituteevidence that mathematical properties, such as number–space mappings,are indeed hardwired, providing the building blocks for the number line?It would. Except that the evidence reported in those studies is not aboutspontaneous number (or numerosity)–to–space mappings (Núñez & Fias,2015). The study withmonkeys (Drucker&Brannon, 2014) investigates aparticular instance of a space-to-space (not numerosity-to-space) mappingfollowing training. And the studies with infants (deHevia&Spelke, 2009,2010) and newborn chicks (Rugani et al., 2015) do investigate numerositybut do not provide evidence of mappings. Rather, at best, they show thatthere might be associations between (or biases in) numerosity and space.Once again, as with the case of the number–numerosity distinction, weneed conceptual and terminological clarity in order to avoid confusionwhen addressing the phenomena under investigation. The notion of map-ping is essential for investigating the number-line concept, so we musthave a clear definition of it.

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When applied to numbers, a standard definition of mapping, as itappears in the Encyclopædia Britannica (2005), is: “any prescribed way ofassigning to each object in one set a particular object in another (or thesame) set. Mapping applies to any set: a collection of objects, such as allwhole numbers, all the points on a line, or all those inside a circle.” Andthe notion of mapping can be used to characterize the number-line con-cept with two defining criteria (Núñez, Cooperrider, &Wassmann, 2012):

(i) there is an actual mapping of each individual number (ornumerosity) under consideration (e.g., counting numbers withina certain range) onto a specific location on the line, and that

(ii) this mapping defines a uni-dimensional space representing num-bers (or numerosities) with a metric (at least approximative)—adistance function.

That the mapping defines a space with a metric means, in the case ofa straight line, that it constitutes a one-dimensional space (i.e., a line)representing numbers with a translation-invariant Euclidean metric:the length of the segment spatially representing the difference of twonumbers satisfies the properties of a Euclidean distance function in aone-dimensional space, which is invariant under addition (if the mappingis logarithmic, these properties apply on a log-transformed space). Thenumber line, whether linear or logarithmic, thus has essential mappingproperties that go well beyond the numerosity–space associations reportedfor preschool children and infants (de Hevia & Spelke, 2009, 2010)or for newborn chicks (Rugani et al., 2015). Such associations havebeen interpreted as evidence of a number-to-line “mapping,” yet theydo not assign to each number (or numerosity) in the collection underconsideration a particular location in space (criterion i) and thus do notconstitute actual number-to-space mappings, let alone define a space witha metric (criterion ii). Indeed, the reported number–space associationsin preschool children (de Hevia & Spelke, 2009) were obtained on thebasis of a bisection task in which participants were asked to indicate themidpoint of a line segment flanked by the numbers (or numerosities) “2”and “9.” In that study a number–space association was said to exist ifthe mean reported midpoint exhibited a slight bias toward the number(or numerosity) “9.” The reported associations for infants (de Hevia &

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Spelke, 2010) were obtained by means of a looking habituation paradigm.A number–space association was said to exist if infants transferred thediscrimination of ordered series of numerosities to the discrimination ofan ordered series of line segments. The associations reported for newbornchicks (Rugani et al., 2015) were based on a left–right binary explorationchoice. After being trained with a target numerosity (5 elements for somechicks, 20 for others), the chicks explored an environment containingtwo panels—to the left and to the right, displaying identical numerositieseither smaller or greater than the target (2 or 8 elements, and 8 or32, respectively). An association (interpreted as “mapping”) was saidto exist given that around 70% of the time the chicks preferred theleft panel when the numerosity was smaller than the target and theright one when it was greater. (Importantly, the relative contrast with8 elements—greater than target 5 but smaller than target 20—was nottested within individuals but between groups).

Despite the use of number–space “mapping” in the titles and argu-mentation of these reports, no actual number (or numerosity)–to–linemappings were investigated therein. The reported associations simplydo not conform to standard definitions of mapping. The phenomenainvestigated in those reports are, at best, biases or associations, whichindeed may manifest quite spontaneously in infants, and they may provideearly cognitive preconditions (not precursors!) for a later consolidationof a mental number line (if necessary cultural conditions are provided).But “mappings” they are not. Mappings, which are essential for theconstitution of the number-line concept, may very well build on theseearly associations—like a snowboarder builds on his/her early balance andlocomotion abilities—but they seem to require more than pure biologicalendowment.

WHAT CAN WE LEARN FROM THE HISTORYOF MATHEMATICS? A LOT

If we want to understand the origin of the concept of number line, animportant source of information is found in the history of mathematics.Here I review two relevant sources, Old Babylonia and the mathematicsof the 17th century in Europe.

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Numbers and Calculations without Number Lines in OldBabylonian Mathematics

If the mental number line is as fundamental and “hardwired” as claimed,we should expect ubiquitous manifestations of number lines throughouthuman history. Leaving aside the fact that far from all human groups haveever developed numerical and arithmetic systems (most never did!), letalone left behind observable evidence of the use of number lines, we canevaluate the innate mental number line claim being maximally permis-sive with regard to the hypothesis and consider exclusively civilizationsknown to have developed sophisticated arithmetical knowledge, such asMesopotamia, ancient Egypt, and China. At least in these old civiliza-tions the evidence of the use of number lines should be overwhelming, asit is the case in our modern society where we see number lines in rulers,calendars, and electronic devices. But no such corresponding evidenceseems to exist, not even in these mathematically sophisticated civiliza-tions. We know from ancient clay tablets in Mesopotamia, for example,that Old Babylonians developed highly elaborated knowledge of arith-metical bases, fractions, and operations without the slightest reference tonumber lines (Núñez, 2008). There are roughly half a million publishedcuneiform tablets, from which no more than 5,000 tablets contain math-ematical knowledge. Only about 50 tablets have diagrams on them (Rob-son, 2008), but none provides evidence of number lines. As the historianof Mesopotamian mathematics Eleanor Robson (written personal com-munication, March 10, 2009) puts it, “There [were] no representations ofnumber lines [in Babylonia]: that metaphor was not part of the Babylonianrepertoire of mathematical cognitive techniques. Until very late indeed(3rd century BC) number was conceptualized essentially as an adjectivalproperty of a collection or of a measured object.” Some of the diagrams onthe tablets, as in the famous YBC 7289 tablet, written around the first thirdof the second millennium BC (see Figure 3.1), do show lines with num-bers associated with measurements. But contrary to what has been claimed(Izard, Dehaene, Pica, & Spelke, 2008), this does not imply that the mak-ers and users of such tablets operated with a number-line concept. Indeed,historians of mathematics have cautiously pointed out the risks of provid-ing “inventive” interpretations of old mathematical documents throughthe lenses of modern concepts (Fowler, 1999) and have wisely suggested

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Figure 3.1 A clay tablet known as YBC 7289 from the Old Babylonian period, fromthe Yale Babylonian Collection. The tablet, from the first third of the second millenniumBC, is one of the few containing drawings. It shows numbers associated with measure-ments but no depiction of number lines. According to contemporary historians ofMesopotamian mathematics, Old Babylonians did not operate with number-lineconcepts despite manifesting sophisticated notions of number and arithmetic.

Source: © Bill Casselman. Reprinted with permission from http://www.math.ubc.ca/people/faculty/cass/Euclid/ybc/ybc.html

that the interpretation of such documents must be made in their historicalcontext (Fowler & Robson, 1998; Robson, 2002).

The tablet YBC 7289 shows a square with its diagonals, the number 30on one side, and the numbers 1, 24, 51, 10 and 42, 25, 35 written againstone diagonal. For decades scholars have interpreted this pair of numbersto be, respectively, a four-sexagesimal-place approximation of

√2 and

the length of the diagonal, and have taken this as an “anachronisticanticipation of the supposed Greek obsession with irrationality andincommensurability” (Robson, 2008, p. 110). But while analyzing

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this tablet in context with other documents, known practices, andarchaeological findings, Robson and colleagues have observed that thetext is most likely a school exercise by a trainee scribe who got theapproximations from a reference list of coefficients (Fowler & Robson,1998). The round shape of the tablet, typically used by trainee scribes atthat time, and the unusually large handwriting (characteristic of trainees)support the idea that the text was not written by a competent scribebut by a student. As part of a series of drills, the trainee scribe simplyfound the length of the diagonal by multiplying 30 (the side) by thepregiven constant 1, 24, 51, 10, exactly as indicated in a well-knowncoefficient list of the time, the tablet YBC 7243 (Robson, 2008). In fact,“there is no evidence that Old Babylonian scribes had any concept ofirrationality” (p. 110). Similarly, another tablet—Plimpton 322, the mostfamous Babylonian mathematical artifact—was long thought to havebeen a trigonometric table. But recent detailed analyses of related tabletsshow that Old Babylonians did not operate with the concept of radius forcalculating areas (they used A = c2/4π, where c is the perimeter, insteadof A = πr2). Therefore, there was no conceptual framework for measuredangle or trigonometry: Plimpton 322 cannot have been a trigonometrictable (Robson, 2002). Tablets such as YBC 7289 and Plimpton 322are nonnarrative and have pictorial or tabular components that can beinadequately read “as ‘pure,’ abstract mathematics, enabling them to berepresented as artifacts familiar to [modern] mathematicians” (Robson,2008, p. 288).

In sum, in the absence of a clear number-line depiction and narrative,simply because we see numerals, magnitudes, and lines on clay tablets,we cannot anachronistically infer that Old Babylonians operated witha number-line mapping or with a mental number-line representation.If, as experts say, Babylonians conceptualized number essentially as anadjectival property of a collection or of a measured object, we cannotconclude on the basis of YBC 7289 that Old Babylonians operated witha fundamental number-to-line mapping. In a nutshell, just because weobserve people making adjectival statements like “plastic chair” and“wooden table,” we cannot conclude that they operate with a fundamentalmaterial-to-furniture mapping. Explicit characterizations of the numberline seem to have emerged many centuries later in Europe, as late as the17th century, and only in the minds of a few pioneering mathematicians.

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How Long Does It Take for the Number Line to Be Invented?A Long Time

According to the known historical record, it was apparently John Wallisin 1685 who, for the first time, introduced the concept of numberline for operational purposes in his Treatise of Algebra (see Figure 3.2).

Figure 3.2 The introduction of the number line in 17th-century Europe. The figureshows the top of page 265 of John Wallis’s Treatise of Algebra, published in 1685, chapter66, “Of Negative Squares and Their Imaginary Roots.” This passage seems to be the firstexplicit characterization of a number line for operational purposes in the history ofmathematics.

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There he begins by introducing a metaphor in which a man movesforward and backward a certain number of yards and establishes basicarithmetical operations where numbers are conceived in terms of motionalong a path. To an educated person alive today, the language and theexplanations used seem strikingly childish and simplistic, akin to how thenumber line is taught today in elementary schools. In any case, Wallismay have built on earlier precursors who paved the way, such as JohnNapier, who, with his 1616 diagrams (originally published in Latin in1614) used an explicit number-to-line mapping to define the concept oflogarithm (see Figure 3.3).

Figure 3.3 The first page of chapter 1 of John Napier’s A Description of the AdmirableTable of Logarihmes published in 1616 (English edition), in which he introduces hisdefinition of logarithm via an explicitly depicted number-to-line mapping.

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Figure 3.4 A diagram from page 392 of Descartes’s first edition of La Géometrie,published in 1637. Contrary to what has been claimed, Descartes did not introduce thenumber line in this classic volume through the invention of coordinate systems.

The number-to-line mapping, however, was not a common idea amongmathematicians. Contrary to what has been claimed (Izard et al., 2008),Descartes did not introduce the number line through the invention ofcoordinate systems in his 1637 LaGéometrie (1637/1954). Descartes’s orig-inal 120-page text never mentions the number-line concept, and none ofits 49 illustrations depicts a number line or a numerical coordinate system,not even when specific values for specific magnitudes are characterized.Figure 3.4 shows one of those illustrations, where the relevant magnitudes“a” and “q,” which could have been displayed along a single encompass-ing left-to-right horizontal number line—as we would do today—are infact depicted separately and vertically. Similarly, other such diagrams inDescartes’s text, displaying up to four different magnitudes, depict theextensions separately, never along the same line. Descartes’s classic textdoes not contain any specific numbers mapped onto lines but only numer-ically unspecified magnitudes in the tradition of Greek geometry.

It is important to point out thatWallis’s and Napier’s texts, intended forreaders with advanced knowledge in mathematics, proceed with detailedand careful—almost redundant—explanations of how to construct and usea number-line mapping. These explanations are not “formalizations” ofthe idea of a number line, but rather, they are elaborated presentations

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0 1 2 3 4 5 6 7 8 9 10

Figure 3.5 An illustration of the number line from a manual from the NationalCouncil of Teachers of Mathematics (in the United States) intended for children inkindergarten to fifth grade. The manual includes practice questions, such as “If I take ahop of 5 and then a hop of 4, where will I land?”

Source: http://illuminations.nctm.org/Lesson.aspx?id=355, Reprinted with permission.

of a new meaningful and fruitful idea that emerges from the demands forgrounding abstract concepts that had turned too elusive—negative squaresfor Wallis and logarithms for Napier. The hand-holding narrative, how-ever, is similar to what we see in many elementary school classrooms today(see Figure 3.5), showing just how unfamiliar the idea of a number linewas to the elite 17th-century mathematicians, let alone to the rest of themajority of illiterate citizens in Europe at that time.

Taken together, these facts from the history of mathematics—from OldBabylonia to 17th-century Europe—are simply at odds with the idea of a“hardwired” mental number line that would spontaneously manifest itselfin all humans. Of course, these facts by themselves do not prove thatthere was no mental number line before the 17th century, but they makethe claim of an innate and spontaneous number-to-line mapping highlyimplausible. Moreover, supporters of such nativist claim must explain:(1) how Babylonian, Greek, and early Renaissance mathematiciansin Europe built so much mathematics without a number-line concept;(2) why it took so long for professional mathematicians (let aloneordinary people) to come up with a notion of number line; and, crucially,(3) why the la crème de la crème of 17th-century European mathemati-cians had to go into such simplistic, childlike language to explain how tomake use of it.

We now turn from these episodes in the history of mathematics, tocontemporary experimental studies in number-to-space mappings, whichhave produced results that are consistent with the historical records.

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ARE RESULTS IN EXPERIMENTAL STUDIESON NUMBER-LINE MAPPINGS CONSISTENTWITH HISTORICAL RECORDS? YES

During the last decade or so, a significant number of studies have beencarried out investigating children’s mappings of numerical estimationsonto lines. Developmental research has shown that when asked to locatenumbers on a line segment marked with a 0 on the left and 100 onthe right—the number-line task—even kindergarteners seem to readilyplace smaller numbers at left of the segment and greater numbers atright (Booth & Siegler, 2006; Siegler & Booth, 2004). Interestingly,they allocate more space to small numbers and less to big numbers in anonlinear—logarithmic-like—compressed manner. The data support theview that numerical estimation follows the domain-general psychophysi-cal Weber–Fechner law that subjective sensation increases proportionalto the logarithm of the stimulus intensity. With schooling and math-ematical training, the development of mapping patterns starts to shiftgradually, between kindergarten and fourth grade, from a logarithmic-likepattern to a primarily linear one (Booth & Siegler, 2006; Siegler & Booth,2004). Based on the assumption that number mappings are indicatorsof mental representations (Dehaene et al., 2008a; Priftis et al., 2006;Zorzi et al., 2002), variations of this number-line task have been usedto investigate the properties of number and numerosity representations(e.g., linear mappings versus logarithmic mappings) in children of variousages and in populations from isolated cultures. In this section we reviewresearch that uses the number line task and addresses two main questionsrelevant to the theme of this chapter:

1. Are number mental representations (numerical magnitude) inher-ently spatial?

2. Is the intuition of mapping number to space hardwired anduniversal?

Are Number Mental Representations Inherently Spatial?

The number-line task has proven to be very useful for the investigationof numerical estimations and number representations in children (andadults). However, little attention has been paid to the fact that the results

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have been obtained with participants reportingmagnitude on a line, whichis an inherently spatial medium. It seems straightforward that a thoroughinvestigation of the nature of number representation must disentangle thenumber–space confound intrinsic to number-line methods, dissociatingnumber stimulus from reporting condition.

More than half a century ago, classic work in psychophysics docu-mented nonlinear compressions of nonspatial stimulus sensory scales thatincluded magnitude production (Stevens & Mack, 1959) and revealeda systematic relationship between them and numerical categories(Stevens, Mack, & Stevens, 1960). Number cognition research, however,obsessed with number-line associations, has largely neglected the studyof nonspatial mappings (with rare exceptions, as in Vierck and Kiesel,2010). Moreover, it has expected spatial representation to be de factoinvolved in number-related experimental tasks, even when the datasuggest no involvement of spatial representations (see Fischer, 2006, fora discussion). Moving in a different direction, recent work in cognitiveneuroscience has advanced the idea that number representation maybuild on a more fundamental magnitude mechanism (Cohen Kadosh &Walsh, 2008; Walsh, 2003), which can be studied nonspatially and mayprovide new insights into the understanding of the nature of numberrepresentation.

In a recent experimental study, Núñez, Doan, and Nikoulina (2011)investigated number mappings using spatial and nonspatial reportingmodes in college-level-educated young adults. This population, whichhas already experienced the reported shift from logarithmic to linearmappings, exhibit differential linear-log mappings that vary according tostimulus modalities (Dehaene et al., 2008a). That is, when reporting ona line, educated adults exhibit logarithmic mappings when respondingto number stimuli that are difficult to count or discern (e.g., tones andlarge numbers of dots) but produce linear mappings when responding tosymbolic stimuli (e.g., words) or to small nonsymbolic number stimulipresented visually (e.g., small numbers of dots). In this study, Núñezand colleagues reasoned that if number representation builds not juston spatial resources but on deeper magnitude mechanisms, one shouldexpect these differential mappings to be similarly enacted with nonspatialreporting conditions. The study, including symbolic and nonsymbolic

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number/numerosity stimuli modalities, considered three nonspatialreporting conditions: two manual–instrumental ones where participantssqueezed a dynamometer and struck a bell with various intensities,and one noninstrumental condition, where they vocalized with variousintensities.

The study replicated previous results with educated participants whohad reported spatially (Dehaene et al., 2008a): nonlinear compression forhard-to-count nonsymbolic stimuli (Figures 3.6N and 3.6O), and linearmappings for easy-to-count 1–10 dots and symbolic stimuli (Figures 3.6M,3.6P). But the authors found that when number estimations werereported nonspatially, the mappings elicited by nonsymbolic stimuli were

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Source: Reprinted from Núñez et al., 2011, with permission.

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consistently nonlinear throughout (Figures 3.6A–C; 3.6E–G; 3.6I–K).When reporting nonspatially, linearity only occurred as a response tosymbolic stimuli (words; Figures 3.6D, 3.6H, 3.6L). Crucially, whilelogarithmic-like mappings were manifested consistently across all non-symbolic stimulus modalities, spatial reporting produced linear mappingsin the key case of 1–10 dots (Figure 3.6M).

Current proposals that attempt to explain the logarithmic-to-linearshift in people with scholastic training (Dehaene et al., 2008a) cannotexplain these results. One proposal says that logarithmic representationsgradually vanish and become linear with education but that they remain“dormant” for very large numbers or whenever there is number approx-imation. A second proposal states that experience with measurementand with the invariance principles of addition and subtraction—essentialfeatures of number concepts—may play an important role in the shift. Butthe results of the study do not support these arguments. First, logarithmicthinking in educated participants, far from being “dormant” or vanishing,did manifest consistently in all nonspatial reporting conditions— instru-mental and noninstrumental. It is worth pointing out that peripheralsensorial compression, which account for magnitude effects in sensoryneurons (Nieder & Miller, 2003), cannot explain by itself the nonlin-earity patterns observed with nonspatial reporting, since these reportingconditions systematically produced linear responses for symbolic stimuli(words). Second, when reporting nonspatially, college students—whoare intensely exposed to measurement practices as well as to additionand subtraction—did not map nonsymbolically presented numerositiesin a linear manner. When reporting nonspatially, they even failed toproduce a linear mapping in the simple 1–10 dots case. The invarianceprinciple, which profiles the equality of numerical magnitude betweenpredecessors and successors of a given natural number (e.g., the absolutedifference between 6 and 5 is the same as the absolute difference between6 and 7), seems to apply only when participants report on a line, wherelinear mappings are produced. But reports via squeezing, bell-striking, orvocalizing produced logarithmically compressed mappings that do nothold invariance since allocated reporting magnitude units were larger forsmall number stimuli than for big ones. It is as if the difference between“3” and “2” and the difference between “9” and “8” are judged as equal

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when reporting on a line but not when reporting nonspatially where theformer difference is reported to be larger.

In the Núñez et al. (2011) study, the only case of linearity manifestedin 1–10 dots was observed with spatial reporting (Figure 3.6M). This caseof linear mapping, although widely studied, appears to be an exceptionrather than the norm. This particular case may indeed be explained byexposure to measurements—a culturally shaped activity through which afixed unit is applied to different spatial locations (thus implementing andembodying the invariance principle). Since logarithmic mappings are pre-ferred by children (Booth & Siegler, 2006; Siegler & Booth, 2004), thenonlinear representations of nonsymbolic quantities seem to be—even inthe college-educatedmind—not just “dormant” but themost fundamentalones. The details of neural organization in the primate brain may provide(at least partial) support for logarithmic encoding. For instance, behav-ioral and neuronal representations of numerosity in the prefrontal cortexof rhesus monkeys exhibit a nonlinearly compressed scaling of numeri-cal information as characterized by the Weber–Fechner law and Stevens’spower law for psychophysical magnitudes (Nieder & Miller, 2003). Thisis consistent with the fact that in this study, numerical estimation bycollege-educated participants followed these laws only when the stim-uli were primarily nonsymbolic (numerosities), but not when they weresymbolic (number words). The monkey results show that the compressedscale is the natural way that numerosity is encoded in the primate brain.It has been argued that this is the case in the language-free monkey brain(Dehaene et al., 2003). But the results in the Núñez et al. study (2011)suggest that, when nonsymbolic stimuli (numerosities) are concerned, thismight be the case even in the human college-educated brainwith language.The differential linear/nonlinear patterns produced by symbolic and non-symbolic stimuli make it unlikely that number/numerosity representationis determined uniquely by the internal organization of cortical represen-tations (Dehaene et al., 2003), requiring the participation of other neuralpathways and brain dynamics involved with culturally mediated symbolicand language processing (Ansari, 2008).

In the studybyNúñezandcolleagues (2011), linearmappingsmanifestedonly in specific culturally supported cases: symbols (words) and small quan-tities of dots when they were reported on a notational device—the linesegment. The case of 1–10 dots is particularly revealing, for it picks out

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four essential features present in the culturally mediated learning of thepowerful concept of the number line: (i) nonsymbolic material, presentedin (ii) small quantities of (iii) visually perceivable (iv) discrete objects. Theinitial exposure to and learning of the number line in early school years isdone, precisely, in the presence of these essential features: bymapping ontothe horizontal line quantities representing discrete small amounts of visu-ally perceivable items, such as single-digit counts of hops (Ernest, 1985;and see Figure 3.5). The remaining stimulus modalities lack these four fea-tures: 10–100 dots may be visually perceivable, but they are not readilycountable or discernible; tones are not perceived visually; and words aresymbolic. It is a fact of our cultural educational practices that the learningof the number line is not donewith tones or with large numbers.Moreover,learners are never (or hardly) exposed tomapping numbers ontomodalitiessuch as squeezing, bell-striking, and vocalizing. This lack of exposure andexperience is reflected in the levels of imprecision of nonspatial responsesin theNúñez et al. studywhose variability across participants was systemat-ically higher than in the ubiquitously trained spatial one. Thus, culturallymediated symbolic-linguistic dimensions may account for the only linearmappings observed in this study. That is, they account for the specificityof linear reports which were exclusively given in response to culturallycreated symbols (number words) across all reporting conditions —spatialand nonspatial—and to 1–10 dots when reported on a culturally creatednotational device: the line segment.

Now, if the linear properties of number mappings do not seem to bethe most fundamental ones, why is the number-to-line mapping so power-ful and ubiquitous? The reason seems to be that the spatial medium offersunique affordances and advantages over othermedia.When usingmarkingdevices, number-to-space reporting can be stored, readily shared with oth-ers, and inspected days, months, or years later. Moreover, building on thespecificities of the primate’s body morphology and brain, which is largelydevoted to visual processing and object manipulation, the spatial mediumallows for close visuomanual monitoring, precision, and metacognitiveprocessing. These advantageous features—absent, or weak, in nonspatialreporting—seemtohavebeenculturally selectedandprivileged, eventuallyleadingtotheelaborationofnumber-linemappingsmediatedvianotationalandmeasuringdevices (Ifrah, 1994) and to the complexities of number-lineand graphic displays that are ubiquitously present in the modern world.

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In sum, the answer to the main question in this subsection—“AreNumber Mental Representations Inherently Spatial?”—seems to be“No.” The results reviewed here suggest that number representation is notinherently spatial but rather that it builds on a deeper magnitude sensethat manifests spatially and nonspatially mediated by number/numerositymagnitude, stimulus modality, and reporting condition. Number-to-spacemappings—although ubiquitous in the modern world—do not seem tobe part of the human biological endowment but have been culturallyprivileged and developed.

Is the Intuition of Mapping Number to Space “Hardwired”and Universal?

The majority of the research in human number cognition has been car-ried out with educated participants, mostly from cultures with writingsystems that make extensive use of number-to-space mappings. Since lit-tle (or almost nothing) is known about the psychology and neuroscienceof number mappings in societies with no writing and measurement prac-tices, it is highly likely that the theoretical accounts we have today arebased on a biased sample. Hence the importance of investigating such pop-ulations before their cultural and language traditions are absorbed by theever-increasing world globalization. In this section we review two suchstudies, one conducted with the Mundurukú from the Amazon (Dehaeneet al., 2008a) and one with the Yupno of the remote mountains of PapuaNew Guinea (Núñez et al., 2012).

Unintended Results with the Mundurukú of the Amazon:Failing to Exhibit Number-Line Mappings

The Amazonian Mundurukú is a group known for having a languagewith a reduced lexicon for precise numbers—1 through about 5 only(Pica, Lemer, Izard, & Dehaene, 2004; Strömer, 1932)—and littleexposure to education and measuring devices. Since the Mundurukúcan operate with sophisticated quantity and spatial concepts in anapproximate and nonverbal manner (Dehaene, Izard, Pica, & Spelke,2006; Pica et al., 2004), Dehaene and collaborators adapted Siegler andcolleagues’ number-line task (Booth & Siegler, 2006; Siegler & Booth,2004; Siegler & Opfer, 2003) for their purposes using symbolic (words)

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and nonsymbolic stimuli (dots and tones) in the 1–10 range (Dehaeneet al., 2008a). Stimulus numbers “1” and “10” were used and presented asanchors at the left end and the right end of the line segment, respectively.After running the task, the authors reported that “the Mundurukú’smean responses revealed that they understood the task. Although someparticipants tended to use only the end points of the scale, most usedthe full response continuum and adopted a consistent strategy of map-ping consecutive numbers onto consecutive locations” (p. 1217). Theauthors further reported that Mundurukú mapped symbolic (words) andnonsymbolic (dots, tones) numerosities onto a logarithmic scale, whileWestern control participants used linear mapping with small or symbolicnumbers and logarithmic mapping when numerosities were presentednonsymbolically. They concluded that (a) the concept of a linear numberline is a product of culture and formal education and (b) the mapping ofnumbers onto space (with metric) is a universal intuition and this initialintuition of number is logarithmic. Beyond the reported logarithmiccompressions of participants’ judgments, which may be explained in lightof the psychophysical Weber–Fechner law, the data seem to support (a),which is consistent with the findings in the study (Núñez et al., 2011)analyzed in the previous subsection. But, ironically, a careful analysisof the Mundurukú study shows that the data in fact do not support themost important claim (b)—that the mapping of numbers onto space is auniversal intuition. What is more, these data support the very oppositeclaim: that the mapping of numbers onto space (with metric) is not auniversal intuition. How is this possible?

The Supporting Online Material of the Mundurukú study mentions(but, surprisingly, does not analyze) that some participants tendedto produce “bimodal” responses using only the endpoints of the linesegment—not the full extent of the response continuum (Dehaene et al.,2008b). According to Dehaene et al.’s own words, the number-line taskhas a fundamental criterion that “participants evaluate the size of thenumbers and place them at spatial distances relative to the endpoints thatare proportional to their psychological distances from those endpoints”(Dehaene, Izard, Pica, & Spelke, 2009, p. 38c). Bimodal responses,however, are categorical responses that violate this fundamental criterionsince they do not reflect psychological distances with a metric—the“distance function” of criterion (ii) mentioned in p. 91—and therefore

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Figure 3.7 Detail from data reported in Dehaene et al. (2008a, 2008b) showingthe mean response location on the line segment that participants picked ascorresponding to the numerosities 1, 2, and 3—numbers for which Mundurukú speakershave a well-established lexicon. Data are mean ± standard error of the mean. The toprow (a) shows values for 16 American participants; the middle row (b) shows values for33 Mundurukú participants; and the bottom row (c) shows values for 7 Mundurukúuneducated adults.(Cont.)

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Figure 3.7 (Cont.) If participants understand the task and spontaneously mapnumbers to the line, they should systematically map the lowest stimulus number (“1”)with the response location “1”—that is, with the left end of the line segment presentedto them. American participants (a) accurately and systematically mapped “stimulusnumber 1” with “response location 1.” Mundurukú participants, however, systematicallyfailed to establish this fundamental number–space mapping (b). Crucially, data reportedexclusively in the corresponding “Supporting Online Material” (Dehaene et al., 2008b)show that the Mundurukú uneducated adults—the most relevant subgroup for testing theinnate mental number line hypothesis—failed to do this in a more dramatic way (c). Forwords and tones, this group even failed to establish the fundamental property of order.These results suggest that the number-line intuition isnot universal.Source: From Núñez, 2011, replotted from Dehaene et al., 2008a (Figures 3.7a and 3.7b) and Dehaeneet al., 2008b (Figure 3.7c), with permission from theAmerican Association for the Advancement ofScience.

cannot be interpreted as number-line mappings proper. Moreover, whentheir frequency is considerably high, they demand further analyses.Strikingly, in the Mundurukú study, 13 experimental runs out of 35(37%) were labeled as bimodal (Dehaene et al., 2008b)—a very highpercentage considering the claim that the intuition of the number-linemapping is universally spontaneous. Indeed, according to the universalityclaim, no runs at all should be expected to be bimodal. In the Mundurukústudy, however, even if the universal hypothesis is considerably loosenedto accept as many as 20% of the runs to be bimodal, the observedfrequency of the reported bimodal responses is still statistically significant(𝜒2= 6.43, df = 1, p = 0.011). This high proportion of bimodal responsesis simply at odds with the conclusion that indigenous people withoutinstruction spontaneously map numbers onto a line.

Furthermore, the high proportion of bimodal responses helps explainanother important problem in the data: the reported Mundurukú “meanresponse locations” do not corroborate that the Amazonian participantsdid establish the lowest-number-to-endpoint mapping—an essential com-ponent of the number-to-space mapping as specified by the number-linetask (Siegler & Booth, 2004; Siegler & Opfer, 2003). A detailed analysisof the data shows that the Mundurukú—especially the unschooledones, which is the group that matters the most for testing universalclaims—failed to establish the basic “numerical anchors” (Booth &Siegler, 2006). Figure 3.7 shows that while the mean response locations

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of control (American) participants accurately and systematically mapped“stimulus number 1” with “response location 1”—the left end point ofthe segment (Figure 3.7a), the values for the Mundurukú participantsshow that they routinely failed to do so (Figures 3.7b and 3.7c). This iseven more striking as the failure involved the number and numerosities“one,” “two,” and “three,” which are the only cases for which Mundurukúspeakers systematically use specific lexical items to identify them (theonly terms used in more than 70% of the cases in which the correspondingquantity was presented; Pica et al., 2004).

The left graph of Figure 3.7b shows that for word numerals inMundurukú (symbolic stimuli), the mean response location for the small-est stimulus number “1” was approximately “2.5”—that is, 1.5 locationunits to the right of the left endpoint. Moreover, the standard error, beingabout 0.5, indicates that there is a substantial variability in the response,meaning that for some participants, the mapped location for stimulusnumber “1” was even farther away from the left endpoint of the segment.Similarly, the center and right graphs of Figure 3.7b show that for dotsand tones (nonsymbolic stimuli), the fundamental mapping 1-to-leftendpoint was not established either. The mean response location for “onetone” was nearly 2 location units away from the left endpoint— that is,a distance corresponding to nearly 22% of the extension of the segment.Most important, data reported only in the corresponding SupportingOnline Material show that for the Mundurukú uneducated adults—themost relevant subgroup for testing the innateness hypothesis—the failurewas even more telling. The left and right graphs in Figure 3.7c showthat the mean mapped location for the word “one” and for “one tone,”respectively, was roughly 3 location units away from the left endpoint(about 30% of the extension of the segment), with a very high standarderror, indicating that for some participants, the mapped location waseven farther away to the right. Moreover, the uneducated Mundurukúadults even failed, for these cases, to establish the fundamental propertyof order: the mean response location for stimuli numerosities “1,” “2,”and “3” are virtually the same, with “one tone” being even farther to themiddle of the line segment than “two tones” and “three tones.” (Note: Inthis study, the presented line segment had a set of 1 dot on the left and aset of 10 dots on the right constantly present on screen; Dehaene et al.,

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2008a, p. 1218. This may explain the presence of order for the responsesin the “dots” condition [center of Figure 3.7c]. These responses may havebeen driven by a perceptual resemblance between the numerical stimulus[dots] and the presented segment [with dots at the endpoints], a situationthat was not available for the “word” and “tone” conditions.)

Considering the high proportion of bimodal Mundurukú responses,the “mean” responses reported in this study are then largely based onartifactual values obtained from averaging a high proportion of endpointresponses. Therefore, they should not be considered as central tendencymeasurements characterizing actual locations on the extent of the linesegment. Rather, they should be considered as relative proportions of left-to right-endpoint responses. In conclusion, the high bimodal Mundurukúresponses, which are categorical responses that fail to reflect psychologicaldistances with a metric, support, in fact, the opposite claim defendedby the authors: the spontaneous intuition of mapping numbers to linearspace (with a distance function) is not universal.

Number Concepts without Number Lines inthe Yupno of Papua New Guinea

A recent study that used essentially the same methodology as the one inthe Mundurukú study was conducted with the Yupno of the mountainsof Papua New Guinea (Núñez et al., 2012). Unlike the Mundurukú,the Yupno have number concepts and a number lexicon beyond 20,and they have access to a creole (Tok Pisin) with an English-basednumber lexicon. Like other indigenous groups in Papua New Guinea(Saxe, 1981), the Yupno have a body-count system, which establishesa number-to-space mapping that exhibits (local) properties of orderbut that lacks a metric—that is, a distance function mentioned earlierin this chapter. Importantly, however, they lack tools and practices forprecise space or time measurements (Wassmann & Dasen, 1994). Othergroups in Papua New Guinea make use of simple measuring practices,such as using the arms for measuring the depth of string bags (Saxe &Moylan, 1982), but the Yupno do not exhibit such practices. The Yupno,then, provide an excellent cultural group for testing the possibility thatnumber concepts may exist without measurement practices and withoutspontaneous number-to-space mapping intuitions.

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As in the Mundurukú study, Núñez and colleagues (2012) usedsymbolic (words) and nonsymbolic stimuli (dots and tones) in the 1–10range. Because one of the main goals of the study was to test the genuinespontaneity of number-line mappings, the researchers carefully scriptedthe task instructions with specific wordings and gestures intended toavoid any unwitting scaffolding. Moreover, the study considered two setsof instructions. One version (Type-1) which had two number-anchors(1 and 10) and static descriptions. A second version (Type-2)—designedto be more explanatory—had three number-anchors (1, 10, and 5) andlinguistic expressions that used more dynamic descriptions. While Type-1instructions followed the procedure of the Mundurukú study as closely aspossible, Type-2 instructions were added to test whether an explanationmaking overt use of the extent of the path might prompt the number-lineintuition. The study included unschooled Yupno participants and Yupnoparticipants who had a very limited education. To make sure that Yupnosparticipating in the number-line task understood the 1–10 cardinalnumerical lexicon, the researchers designed a simple screening procedurethat explicitly tested their mastery of the cardinal number lexicon in theYupno language. (With very rare exceptions, all participants passed thenumber screening procedure.)

After running the number-line task, the researchers found thatunschooled participants, despite knowing the number terms, had sig-nificant difficulties with the training trials, failing to comprehend thefundamental endpoint anchoring required by the number-line task (seeNúñez et al., 2012, for details). The proportions of failures were signif-icantly higher than those observed for the schooled Yupno participants,who exhibited no endpoint-matching failures. Crucially, the Type-2instructions were not helpful, despite involving dynamic language thatexplicitly showed the mapping of the number 5 onto a location onthe path.

Most important, analyses conducted on blocks that had successfulendpoint anchoring trials (with standard Type-1 instructions) showedthat unschooled Yupnos produced a mapping that systematically ignoredthe extent of the path. In all three stimulus modalities, they exhibited ametric-free bicategorical mapping, where small numbers and numerosi-ties (1, and sometimes 2 and 3) mapped onto the left endpoint and

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Figure 3.8 Pointing responses for stimulus numerosities 3 and 6 (dots) during thenumber-line task.Source: From Núñez et al., 2012.

midsize and large numbers and numerosities onto the right endpoint (seeFigure 3.8 for examples).

The bicategorical mappings produced by unschooled Yupnos—with nouse of the extent of the segment when mapping intermediate numbers andnumerosities—are in stark contrast to those of schooled Yupnos and Cali-fornia controls (Figure 3.9). The high proportions of endpoint responses byunschooled Yupno were extremely significant in all three stimulus modali-ties. And, while schooled Yupnos used the extent of the segment accordingto known standards (Dehaene et al., 2008a; Siegler & Booth, 2004), theydid so not as smoothly as California controls, exhibiting a bias toward theendpoints. This schooled Yupno response pattern suggests an intermediatemastery of the number line that is culturally modulated, in which the pri-macy of the basic bicategorical intuition coexists with the learned distancefunction of the number-line mapping.

The Yupno bicategorical responses appear to be of the same typeas the frequent “bimodal” responses mentioned (but ignored and leftunanalyzed) in the Mundurukú study. The modes of the distributionsare, in both cases, concentrated at the endpoints, with a significant

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40%

20%

1 2 3 4 5

F(9, 36) = 55.17p < 10–15

6 7 8 9 10

0%

1 2 3

segment section4 5

F(9, 45) = 1.28n.s.

6 7 8 9 10 1 2 3 4 5

F(9, 81) = 1.04n.s.

6 7 8 9 10

Figure 3.9 Responses to three different types of number stimuli on the number-linetask. Graphs show proportions of responses on segment sections with correspondingrepeated measures ANOVA statistics. In stark contrast with schooled Yupnos (middlecolumn) and California controls (right column), unschooled Yupnos (left column)did not use the extent of the segment when mapping intermediate numbers andnumerosities, producing instead a bicategorical mapping without a metric (a distancefunction). Schooled Yupnos (middle column) did use the line but tended to manifest abias toward the endpoints.

Source: From Núñez et al., 2012.

amount of responses failing to use the extension of the line segmentto map intermediate number and numerosity stimuli. The Mundurukúendpoint responses, although statistically significant, were not as markedas they were among the unschooled Yupno. But this difference couldbe explained by variations in the experimental procedures employedin the two studies. Mundurukú participants may have been helped byperceptual resemblance cues as they were presented, at all times andfor all conditions, with an image of a line that had 1 dot and 10 dotslocated at the left and right end of the segment, respectively. The crucialpoint is that the significant bimodal Mundurukú responses are not simplyoddities that can be ignored. Instead, they constitute evidence of agenuine phenomenon of lack of spontaneous number-line intuitions, aphenomenon that is manifested also in a completely unrelated culturalgroup—the Yupno.

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The bicategorical pattern found in both the Mundurukú and Yupnogroups could be interpreted as a manifestation of a more fundamentalphenomenon described in developmental psychology. In the investiga-tion of categorical and comparative interpretations by children, Smith,Rattermann, and Sera (1988) stated that “young children seem to inter-pret dimensional terms categorically even when the intended meaning,by adult standards, is clearly comparative” (p. 342). And in their research,they found that the categorical and comparative usages of young children(3-year-olds)—for example, high(er) and low(er)—are restricted to thecorresponding poles, and it is only later that children via a richer use ofcomparatives understand quantitative dimensions as one encompassingsystem. Referring to earlier work on semantic incongruity effects in adultsin which adults’ reaction times suggest that they do not automaticallyprocess the logically necessary relation between opposing terms (Banks,1977), Smith et al. conclude that “children’s initial treatment of opposingterms as separate may reflect some fundamental fact about human cog-nition since the treatment appears both developmentally primitive and,in adults, computationally simple” (p. 356). It is possible then that in theMundurukú and Yupno cultures, the pressure for dealing with quantitativedimensions as one encompassing system has not been as intense as it has inWestern civilization, keeping the saliency of the poles much more promi-nent. And, in the case of the Yupno, this would be consistent with the factthat they do not have measuring practices of any sort and, importantly,their language does not have comparatives in the grammar (J. Slotta, per-sonal communication, June 28, 2013) that might help solidify the conceptof quantitative dimensions. Thus during the number-line task, when con-fronted to a notational device never seen before—the line segment—theunschooled participants from these cultural groups may have enacted thisbasic polar construal built around the (opposite) endpoints of the segmentonto which small and big numbers were mapped.

The results from the Yupno and the Mundurukú studies, when takentogether, demonstrate that the intuition of the number-to-line mappingas reflected by the number-line task is not universally spontaneous, andtherefore, it is unlikely to be rooted directly in brain evolution. Althoughubiquitous in the modern world, the number-to-line mapping seems to be

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learned through—and continually reinforced by—specific cultural prac-tices. This conclusion is consistent with the findings in the study on num-ber mappings with nonspatial reports discussed in the previous subsection(Núñez et al., 2011), showing that number-to-space mappings are not asfundamental as previously thought and that linear number-line mappingsrequire cultural practices to be established. Moreover, these observationsalso seem to match the available records of the history of mathematicsdiscussed earlier, which show no documentation of depictions of numberlines proper prior to the 17th century (Núñez, 2009, 2011).

In sum, evidence from several sources—from developmental tocross-cultural—points to a basic human association between number andspace. It is possible that this constitutes an early precondition for theestablishment of number-line concepts. But the results analyzed in thissection suggest that the emergence of the number line proper (i.e., withnumbers mapped onto a unidimensional space with a metric—a distancefunction) requires considerable sociohistoric and cultural mediation,acting outside of natural selection in biological evolution. These basicassociations, therefore, may be preconditions for the number line, but theyare not precursors of it. Specific cultural practices, such as the use of mea-surement tools, notational and graphical devices, writing systems, and theimplementation of systematic education, have served to establish, refine,and sustain the specificities of the number-line mapping. As recent workin the neuroscience of number cognition (Ansari, 2008) suggests, it islikely that, over the course of prolonged and systematic exposure to thesecultural practices in educated individuals, brain areas such as the parietallobes are recruited to support number-line representation and processing.

BIOCULTURAL ISSUES FOR CHILD PSYCHOLOGYAND DEVELOPMENTAL THEORY: IS SNOWBOARDING“HARDWIRED”? NO, IT IS NOT

The view that the number line is “hardwired” is attractive and convenientbut oversimplistic and misleading. It contains teleological componentsthat hide the sociotemporal complexity of human high-order cognitionand its biocultural underpinnings. An analogy may help identify theteleological argumentation of the nativist position. In order to make

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the points more salient, rather than asking the question of the nature ofmathematics and of related basic intuitions such as the number line, letus consider asking the question of the nature of snowboarding. Mathe-matics, like snowboarding, is observed only in humans. But if we think ofa situation in which we would live in a global village where all (or most ofthe) humans we would normally encounter practiced snowboarding as themajor form of locomotion, we would take snowboarding—like the numberline—fully for granted and see it as a “natural” and “spontaneous” activityperformed by (most) healthy individuals. In such a world, when scanningbrains and studying neurological injuries of fellow snowboarders, wewould be led to believe that there are “snowboarding areas” in the humanbrain, and we would see crawling infants as engaging in “protosnow-boarding” and manifesting “early-snowboarding” capacities. Without adetailed, systematic, and in-depth investigation of the brains and behav-iors of human ancestors’ locomotion (even outside of our little snowymountain) and of exotic no-snowboarding humans, we would be easilyled to consider the basic components of snowboarding as “hardwired.”

From the understanding of the real world we inhabit, however,we unmistakably know that snowboarding was not brought forth bybiological evolution—snowboarding is not part of the human biologicalendowment. Clearly, in order to snowboard, we need a brain and makespecific use of neural mechanisms shaped by evolution such as balanceregulation, optic flow navigation, appropriate motor control dynamics,and so on. These mechanisms may constitute early preconditions forsnowboarding, but in themselves, these mechanisms are not precursors ofsnowboarding and are not about snowboarding: they cannot explain theemergence of snowboarding proper. In addition to certain environmentalsettings foreign to human biological evolution (e.g., mountain slopes withsnow), snowboarding requires crucial cultural mediation. For instance,snowboarders must be members of a culture that has already solved prob-lems of thermic insulation of the human body such that they don’t sufferfrom lethal hypothermia when practicing such forms of locomotion. Theymust also be a part of a culture that has invented sophisticated materialsfor optimal board sliding that do not exist naturally in the environmentand lift gondolas such that they can optimize the use of energy allowingfor repeated downhill slides, which avoids the tiring and time-consuming

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climbing back up the hill after each slide. Such technology enables fastand efficient improvement of snowboarding techniques that developin the ontogeny of individuals and the learning of a very unnaturallocomotion pattern, one that clearly did not evolve through biologicalevolution: locomotion with highly restricted lower-limb movementindependence occurring in freezing conditions. And so the analogy canbe fleshed out with any desired level of detail.

The moral is that humans may indeed have evolutionarily driven“hardwired” mechanisms for numerosity judgments (e.g., subitizing, largenumerosity discrimination provided that they differ by specific ratios)and perception of stimulus intensity (e.g., Weber–Fechner law) that mayconstitute early preconditions for numerical and arithmetical abilities.But these mechanisms in themselves do not provide an explanationof the nature of arithmetical or mathematical entities proper, noteven of fundamental ones such as the number line: basic numerositydiscrimination (which is biologically endowed) is to number-to-spacemappings (culturally shaped and learned by individuals) as crawling is(not to walking but) to snowboarding. Evolutionarily shaped mechanismssuch as optic flow navigation and numerosity discrimination can berecruited for the consolidation of behaviors involving snowboardingand the number line, respectively, but in themselves, they do not tellus about the nature of snowboarding or the mental number line. It isthen misleading to teleologically consider young infants engaging incrawling locomotion and manifesting large numerosity discriminationabilities as “early” (or “proto-”) snowboarding or as “early” (or “proto-”)arithmetic, respectively. Arithmetic and the mental number line (as theterm suggests) is about numbers, and numbers, as such, are much morethan numerosity judgments and perceptual discrimination capacities.They are sophisticated human symbolically sustained concepts that,while they build on biological resources and constraints, are culturallyand historically mediated by language, external representations, writingand notation systems, and the need to solve specific societal problems(Núñez, 2009). None of these crucial components that make numbersand the number line possible is “hardwired,” just as thermic insulationand gondola technology in the case of snowboarding is not “hardwired.”

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The claim that the mental number is hardwired can be testeddiachronically and synchronically. If the mental number line is sharedby all humans, we must observe manifestations of it (1) throughout thehistory of humankind and (2) in all cultures around the world today. Butwe don’t. Archaeological and historical data—from Old Babylonia to17th-century Europe—are simply at odds with the idea of a “hardwired”mental number line that would spontaneously manifest itself in allhumans. And unschooled Mundurukú adults dramatically failed to mapeven the simplest numerosity patterns—one, two, and three—with a linesegment, and a high proportion of them used only the segment’s end-points, failing to use the full extent of the response continuum. The samebicategorical response was observed in a completely unrelated culture onthe other side of the globe, with the totality of the unschooled Yupnoparticipants across all stimulus modalities. Establishing the amazinglyefficient number-to-space mapping as the number line is an extraordinaryfact in human history, which took centuries—if not millennia—ofcultural inventions building on strongly constrained human cognitivecapacities. The mental number-line phenomena we observe in peoplethese days seem to be the manifestation of the psychological and neuro-logical realization of today’s well-established number line (and its relatedmental representations) that people acquire through cultural practicesand educational exposure. The currently known behavioral and neu-roimaging data, almost exclusively gathered with schooled participantsfrom the industrialized world, actually reveal how the neural phenotyperealizes the culturally created number line, rather than give support tothe idea that all Homo sapiens map numbers to space in a number-linemanner.

In this chapter I took on the question of how much of mathematicsis “hardwired” (if any!). I addressed it primarily with the relatively sim-ple case of the number line, to conclude that the spontaneous intuitionof number-to-line mappings is not innate, not part of the human biologi-cal endowment, but rather it requires considerable cultural mediation andscaffolding for it to be brought to being. And what about more elaboratedmathematical concepts? Are they “hardwired”? I leave it to the reader tocome up with his or her own conclusion.

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Minnesota Symposia on Child Psychology, Volume 38nunez/web/Nunez_ch3_MN.pdf · 2017-04-13 · functioning in corresponding brain areas was momentarily disrupted (Göbeletal.,2006).Moreover,forseveraldecades,lesionstudieshave